Annals of Fuzzy Mathematics and Informatics
Volume 6, No. 2, (September 2013), pp. 415–424
ISSN: 2093–9310 (print version)
ISSN: 2287–6235 (electronic version)
http://www.afmi.or.kr
@FMI
c Kyung Moon Sa Co.
°
http://www.kyungmoon.com
Decision-making approach with entropy weight
based on intuitionistic fuzzy soft set
Yong Wei Yang, Ting Qian
Received 28 October 2012; Revised 30 December 2012; Accepted 18 January 2013
Abstract.
This paper analyzes the limitations of Jiang et al. approach to intuitionistic fuzzy soft sets based decision making, then a new
entropy measure based on the degree of intuitionistic fuzziness is presented
to calculate the weights of criteria of alternatives. By level soft sets and
score functions, a novel decision-making approach with entropy weight is
proposed. The method considers not only the given condition, but also the
numbers of criteria of alternatives satisfied the given condition. Finally,
examples are given showing that its practicality and effectiveness.
2010 AMS Classification: 03E72, 54A40
Keywords: Intuitionistic fuzzy soft set, Decision making, Entropy, Score function,
Level soft set.
Corresponding Author: Yong Wei Yang ([email protected])
1. Introduction
M
olodtsov [18] introduced the concept of soft sets as a new mathematical tool
to deal with complex systems involving uncertain or not clearly defined objets and
he also pointed out several directions for the applications of soft sets. At present,
soft set theory has received much attention in the field of algebraic structures such
as groups [1], semigroups [3, 20], hemirings [24] and BL-algebras [27]. Jun et al.
applied soft theory to BCK/BCI-algebras [10, 11] and d-algebras[12], respectively,
and investigated their properties.
Since Maji et al. [15] introduced the notion of fuzzy soft sets, as a generalization of the standard soft sets, some researchers have shown great interest in fuzzy
soft sets [4, 25, 26]. Maji et al. presented a fuzzy soft set theoretic approach of
object recognition from an imprecise multi observer data [19] and a neutrosophic
soft set theoretic approach towards the a multiobserver decision making problem
[17], respectively. Based on grey relational analysis, Kong et al. [13] proposed a
Yang et al./Ann. Fuzzy Math. Inform. 6 (2013), No. 2, 415–424
new algorithm whose bases are multiple. Following the research of Feng et al. [6, 7],
Jiang et al. [9] considered level soft sets of intuitionistic fuzzy soft sets, and extended
Feng’s adjustable decision making to the setting of intuitionistic fuzzy soft sets.
In this paper, we first analyzes Jiang et al. approach to intuitionistic fuzzy soft
sets based decision making, then point out its limitations. In order to calculate
the weights of criteria, we present a new entropy measure based on the degree of
fuzziness and intuitionism, then give a example to show its validity. Using the scores
of alternatives at certain level, we propose a novel decision-making approach with
entropy weight based on intuitionistic fuzzy soft sets. The new approach is an adjustable method based on Jiang et al.’s and it considers not only the given condition,
but also the numbers of criteria of alternatives satisfied the given condition. The
new method can be successfully applied in some decision making problems and some
examples are given showing its practicality and effectiveness.
2. Preliminaries
In the section, we will recall some relevant notions which will be used in the paper.
Definition 2.1 ([2]). An intuitionistic fuzzy set (IFS, for short) Ã in X is given by
à = {(x, µÃ (x), νà (x))|x ∈ X}
where µÃ : X → [0, 1] and νà : X → [0, 1] with the condition: 0 ≤ µÃ (x) + νà (x) ≤
1, ∀x ∈ X.
The numbers µÃ (x) and νà (x) represent, respectively, the membership degree
and non-membership degree of the element x ∈ X to the set Ã. We denote the set
of all the IFSs in X as IF(X).
πÃ(x) = 1 − µÃ (x) − νà (x) for any x ∈ X is called the degree of indeterminacy of
x to IFS Ã. It is obvious that 0 ≤ πà (x) ≤ 1, x ∈ X. For the sake of simplicity, we
call α = (µα , να ) intuitionistic fuzzy number (IFN), where µα ∈ [0, 1] and να ∈ [0, 1].
The operations of IFS [2] are defined as follows, for every Ã, B̃ ∈ IF (X)
• à ≤ B̃ if and only if µÃ (x) ≤ µB̃ (x) and νà (x) ≥ νB̃ (x) for all x ∈ X.
• à = B̃ if and only if à ≤ B̃ and à ≥ B̃.
• The complementary of IFS à is Ãc (x) = {(x, νà (x), µÃ (x))|x ∈ X}.
• à ∩ B̃ = {(x, min(µÃ (x), µB̃ (x)), max(νà (x), νB̃ (x)))|x ∈ X}.
• à ∪ B̃ = {(x, max(µÃ (x), µB̃ (x)), min(νà (x), νB̃ (x)))|x ∈ X}.
In the following, we will briefly review some concepts related to soft sets. Through
the whole paper, let U be an initial universe of objects and E be a set of parameters
related to the objects in U . Let P(U ) denote the power set of U and A ⊆ U . Then
Molodtsov defined the notion of soft sets as follows.
Definition 2.2 ([18]). . A pair (F, A) is called a soft set over U where F is a
mapping given by F : A → P(U ).
Through the combination of the theories of intuitionistic fuzzy set and soft set,
Maji et al. introduced the concept of intuitionistic fuzzy soft sets.
Definition 2.3 ([16]). A pair (F, A) is called an intuitionistic fuzzy soft set over U ,
where F is a mapping given by F : A → IF(U ).
416
Yang et al./Ann. Fuzzy Math. Inform. 6 (2013), No. 2, 415–424
Note that the concept of level soft sets in fuzzy soft set theory was initiated by
Feng et al. [6]. Based on the concept of level soft sets of fuzzy soft sets, Jiang et al.
defined level soft sets of intuitionistic fuzzy soft sets as follows.
Definition 2.4 ([9]). Let $ = (F, A) be an intuitionistic fuzzy soft set over the
universe U , where A ⊆ E and E be a set of parameters. For s, t ∈ [0, 1], the
(s, t)-level soft set of $ is a crisp soft set L($; s, t) = (F(s,t) , A) defined by
F(s,t) (ε) = L(F (ε); s, t) = {x ∈ U |µF (ε) (x) ≥ s and νF (ε) (x) ≤ t}
for all ε ∈ A.
Definition 2.5 ([9]). Let $ = (F, A) be an intuitionistic fuzzy soft set over the
universe U , where A ⊆ E and E be a set of parameters. Let λ : A → [0, 1] × [0, 1]
be an intuitionistic fuzzy set in A which is called a threshold intuitionistic fuzzy set.
The level soft set of $ with respect to λ is a crisp soft set L($; λ) = (Fλ , A) defined
by Fλ (ε) = L(F (ε); λ(ε)) = {x ∈ U |µF (ε) (x) ≥ µλ(ε) (x) and νF (ε) (x) ≤ νλ(ε) (x)} for
all ε ∈ A.
For real-life application of intuitionistic fuzzy soft sets based on decision making,
the threshold intuitionistic fuzzy soft set λ is in advance chosen by decision makers.
3. The analysis of Jiang et al. decision making approach
In the section, we first show Jiang et al. approach to intuitionistic fuzzy soft sets
based decision making in [9], and point out its limitations.
Jiang et al. proposed an adjustable approach to intuitionistic fuzzy soft sets based
decision making as follows.
Algorithm 1.
(1) Input the (result) intuitionistic fuzzy soft set $ = (F, A).
(2) Input a threshold intuitionistic fuzzy set λ : A → [0, 1] × [0, 1] for decision
making.
(3) Compute the level soft set L($; λ).
(4) Present the level soft set L($; λ) in tabular form and compute the choice
value ci of xi , ∀i.
(5) The optimal decision is to select xk if ck = maxi ci .
(6) If k has more than one value, then any one of xk may be chosen.
In order to describe the basic idea of the above algorithm, let us consider the
following example. Some of it is quoted from [7, 9].
Example 3.1. Let us consider an intuitionistic fuzzy soft set $ = (F, A) which
describes the “attractiveness of houses” that Mr. X is considering for purchase.
Suppose there are six houses in the domain U = {h1 , h2 , h3 , h4 , h5 , h6 } under consideration, and A = {ε1 , ε2 , ε3 , ε4 , ε5 , ε6 } is a set of decision parameters. The εi (i =
1, 2, 3, 4, 5, 6) stand for the parameters “modern”, “cheap”, “beautiful”, “large”,
“convenient traffic” and “in green surroundings”, respectively. Table 1 gives the
tabular representation of the intuitionistic fuzzy soft set $ = (F, A).
If we choose λ = topbottom$ where topbottom$ : A → [0, 1] × [0, 1] defined by
µtopbottom$ (ε) = max{µF (ε) (x)|x ∈ U } and νtopbottom$ (ε) = min{µF (ε) (x)|x ∈ U }
417
Yang et al./Ann. Fuzzy Math. Inform. 6 (2013), No. 2, 415–424
for all ε ∈ A, then topbottom$ = {(ε1 , 0.8, 0), (ε2 , 0.8, 0.1), (ε3 , 0.8, 0), (ε4 , 0.7, 0.1),
(ε5 , 0.8, 0.1), (ε6 , 0.7, 0.1)}.
By using Algorithm 1, we obtain the level soft set L($; topbottom$ ) with the
choice values with tabular representation as in Table 2.
From Table 2, it is easy to see that the maximum choice value is c5 = c6 = 4,
hence h5 and h6 are the optimal alternatives.
U
h1
h2
h3
h4
h5
h6
ε1
(0.3, 0.5)
(0.8, 0.0)
(0.5, 0.4)
(0.2, 0.7)
(0.7, 0.0)
(0.8, 0.0)
Level
U
h1
h2
h3
h4
h5
h6
Table 1
Intuitionistic fuzzy soft set $ = (F, A)
ε2
ε3
ε4
ε5
(0.6, 0.3) (0.6, 0.3) (0.6, 0.3) (0.3, 0.5)
(0.8, 0.1) (0.8, 0.0) (0.6, 0.2) (0.4, 0.1)
(0.5, 0.4) (0.2, 0.6) (0.2, 0.6) (0.5, 0.3)
(0.2, 0.6) (0.0, 0.9) (0.0, 0.9) (0.2, 0.4)
(0.8, 0.1) (0.8, 0.0) (0.7, 0.1) (0.5, 0.3)
(0.8, 0.1) (0.5, 0.4) (0.7, 0.1) (0.8, 0.1)
soft
ε1
0
1
0
0
0
1
set
ε2
0
1
0
0
1
1
ε6
(0.5, 0.2)
(0.6, 0.1)
(0.2, 0.1)
(0.1, 0.7)
(0.7, 0.1)
(0.5, 0.1)
Table 2
L($; topbottom$ ) with choice values
ε3 ε4 ε5 ε6 Choice value (ci )
0 0 0 0
c1 = 0
1 0 0 0
c2 = 3
0 0 0 0
c3 = 0
0 0 0 0
c4 = 0
1 1 0 1
c5 = 4
0 1 1 0
c6 = 4
From the above example, it can see that level soft sets of intuitionistic fuzzy soft
sets serve as bridges between intuitionistic fuzzy soft sets and crisp soft sets, and the
choice value of an alternative in the level soft set represents the number of criteria
satisfied by the alternatives at certain level of membership degrees. Algorithm 1
usually choose the alternatives which satisfy the condition at utmost, but not the
best one under the situation. There are two optimal alternatives in Example 3.1, but
we don’t know which one is better. Thus, in decision making problems, we should
consider not only the difference of criteria of an alternative, but also the entirety of
criteria of an alternative.
4. A new approach based on entropy weight
In order to propose a new approach to intuitionistic fuzzy soft set based decision
making, we introduce the following notions.
It is well known that for fuzzy sets entropy is a measure of fuzziness, while the
notion of IFSs is a generalization of fuzzy sets, IFSs entropy is expected a measure of
intuitionistic fuzziness [22]. By intuition judgement we realize that some of axiomatic
requirements of an intuitionistic fuzzy entropy ignore the influence of the degree of
indeterminacy. If µÃ (x) = νà (x) = 0 for all x ∈ X, we can not get any information
418
Yang et al./Ann. Fuzzy Math. Inform. 6 (2013), No. 2, 415–424
of the object, the entropy of the IFS depends solely on the intuitionistic component,
since the degree of intuitionistic fuzziness is zero, therefore the entropy value should
be maximum. The greater of the degree of indeterminacy, the greater of the value
of IFSs entropy. Hence, the entropy measure should consider the influence of the
degrees of intuitionistic fuzziness and indeterminacy. Inspired by [28], we can give
a modified definition of intuitionistic fuzzy entropy as follows.
Definition 4.1. A real-valued function H : IF S(X) → [0, 1] is called an entropy
for IFSs, if it satisfies the following axiomatic requirements, let Ã, B̃ ∈ IF(X):
(1) H(Ã) = 0 iff à is a crisp set.
(2) H(Ã) = 1 iff µÃ (x) = νà (x) = 0, πà (x) = 1 for any x ∈ X.
(3) If πà (x) ≥ πB̃ (x) and |µÃ (x) − νà (x)| = |µB̃ (x) − νB̃ (x)| for all x ∈ X, then
H(Ã) ≥ H(B̃).
(4) If |µÃ (x) − νà (x)| ≤ |µB̃ (x) − νB̃ (x)| and πà (x) = πB̃ (x) for all x ∈ X, then
H(Ã) ≥ H(B̃).
(5) H(Ã) = H(Ãc ).
Then according to Definition 4.1, we can give a new formula of entropy measure.
Theorem 4.2. Let à be an intuitionistic fuzzy set in the universe of discourse
X = {x1 , x2 , · · · , xn }, then
Pn 1−(µ (x )−ν (x ))2
HY (Ã) = n1 i=1 1+µÃ (xii )+νà (xii ) .
Ã
Ã
is entropy measure of the intuitionistic fuzzy set à which satisfies the requirement
conditions of Definition 4.1.
Proof. Let Ã, B̃ ∈ IF(X).
(1) Suppose that HY (Ã) = 0. Since
1−(µÃ (x)−νà (x))2
1+µÃ (x)+νà (x)
≥ 0 for all x ∈ X, hence
µÃ (x) = 0, νà (x) = 1 or µÃ (x) = 1, νà (x) = 0, thus à is a crisp set. The
converse is obvious.
1−(µ (x)−ν (x))2
(2) Assume that HY (Ã) = 1, then 1+µÃ (x)+νà (x) = 1 for all x ∈ X, hence
Ã
Ã
µÃ (x) − νà (x) = 0 and µÃ (x) + νà (x) = 0, so µÃ (x) = νà (x) = 0 and
πà (x) = 1. The converse is obvious.
(3) If πà (x) ≥ πB̃ (x) and |µÃ (x) − νà (x)| = |µB̃ (x) − νB̃ (x)| for all x ∈ X, then
µÃ (x) + νà (x) ≤ µB̃ (x) + νB̃ (x), hence
1−(µÃ (x)−νà (x))2
1+µÃ (x)+νà (x)
≥
1−(µB̃ (x)−νB̃ (x))2
1+µB̃ (x)+νB̃ (x) ,
so H(Ã) ≥ H(B̃).
(4) If |µÃ (x) − νà (x)| ≤ |µB̃ (x) − νB̃ (x)| and πà (x) = πB̃ (x) for all x ∈ X, then
1−(µÃ (x)−νà (x))2
1+µÃ (x)+νà (x)
(5) Obviously.
≥
1−(µB̃ (x)−νB̃ (x))2
1+µB̃ (x)+νB̃ (x) ,
so H(Ã) ≥ H(B̃).
¤
To show our entropy measure is effective, we cite some researchers’ entropy measures of the intuitionistic fuzzy set à of n elements as follows.
Szmidt and Kacprzyk [21] defined entropy measure of à as
Pn maxCount(à ∩Ãc )
HS (Ã) = n1 i=1 maxCount(Ãi ∪Ãic ) .
i
i
419
Yang et al./Ann. Fuzzy Math. Inform. 6 (2013), No. 2, 415–424
where Ãi = (xi , µÃ (xi ), νà (xi )) and maxCount(Ãi ) = µÃ (xi ) + πÃ(x)
Li et al. [14] defined a new formula of entropy measure of à as
HL (Ã) =
Pn
i=1 (min(µÃ (xi ),νà (xi )))
Pn
.
i=1 (max(µÃ (xi ),νà (xi )))
Vlachos and Sergiadis [22] defined entropy measure of IFSs which is the extension
of nonprobability entropy for fuzzy sets as
¤
Pn £
µÃ (xi )
νà (xi )
1
HV (Ã) = − nln2
i=1 µÃ (xi )ln µ (xi )+ν (xi ) + νà (xi )ln µ (xi )+ν (xi ) − πà (xi )ln2 .
Ã
Ã
Ã
Ã
Example 4.3. Let Ã1 = {(x, 0.3, 0.0)}, Ã2 = {(x, 0.5, 0.0)}, Ã3 = {(x, 0.6, 0.4)},
Ã4 = {(x, 0.1, 0.8)}, Ã5 = {(x, 0.4, 0.4)}, Ã6 = {(x, 0.5, 0.5)} be intuitionistic fuzzy
sets in the set X = {x}. We calculate the entropies of the IFSs Ãi , i = 1, 2, 3, 4, 5, 6
and obtain the entropy values of IFSs Ãi with tabular representation as in Table 3.
From Table 3, we see that the entropy values of IFSs Ã1 and Ã2 are equal to HL ,
they are contradictory with our intuitive judgment. The entropy values of IFSs Ã3
and Ã4 calculated by HV are very larger than the ones calculated by other formulas,
therefore it is not efficient to measure the entropy of IFS. For Ã5 , µA˜5 = νA˜5 = 0.4
and πA˜5 = 0.2, then the degrees of intuitionistic fuzziness and indeterminacy exist;
for Ã6 , µA˜6 = νA˜6 = 0.5 and πA˜6 = 0, only the degree of intuitionistic fuzziness
exists, while the entropy values calculated by the previous formulas are the same,
thus they can not describe the difference between IFSs Ã5 and Ã6 . Our formula of
entropy measure is very efficient to measure the information of the IFSs Ãi .
i
Ãi
HS (Ãi )
HL (Ãi )
HV (Ãi )
HY (Ãi )
Table 3
Tabular representation of the entropy values of IFSs Ãi
1
2
3
4
5
6
(0.3, 0.0) (0.5, 0.0) (0.6, 0.4) (0.1, 0.8) (0.4, 0.4) (0.5, 0.5)
0.6000
0.5000
0.6667
0.2222
1.0000
1.0000
0
0
0.6667
0.1250
1.0000
1.0000
0.6000
0.5000
0.9710
0.5529
1.0000
1.0000
0.6000
0.5000
0.4800
0.2684
0.5556
0.5000
Since the ordered relation in IFS is not total, and not any two intuitionistic fuzzy
sets are comparable. While score function can solve the problem by converting IFNs
into real numbers. In the following, we define a new score function of IFS with the
influence of the degree of indeterminacy.
Definition 4.4. Let α = (µα , να ) be a IFN, then the score function of α is:
S(α) = µα − να +
2(µα −να )
1+µα +να πα .
By extending some concepts of [5], we introduce some basic notions as follows.
From now on, |A| is the cardinality of A and NA = {1, 2, · · · , |A|}.
Definition 4.5. Let $ = (F, A) be an intuitionistic fuzzy soft set over the universe
U , where A ⊆ E and λ be a threshold intuitionistic fuzzy set. The function δL($;λ) :
U → N is defined by
X
δL($;λ) (x) =
χFλ (ε) (x)
ε∈A
420
Yang et al./Ann. Fuzzy Math. Inform. 6 (2013), No. 2, 415–424
is called the choice value function of $ respect to λ, where χFλ (ε) denotes the
characteristic function of Fλ (ε).
Definition 4.6. Let $ = (F, A) be an intuitionistic fuzzy soft set over the universe
U , where A ⊆ E and λ be a threshold intuitionistic fuzzy set. The choice value
soft set of $ respect to λ is a soft set Γ(L($; λ)) = (κL($;λ) , NA ) over U , where
κL($;λ) : NA → P(U) is given by
κL($;λ) (n) = {x ∈ U |δL($;λ) (x) ≥ n}.
Let’s denote ΥL($;λ) = max{δL($;λ) (x)|x ∈ U } as the choice value rank of the
level soft set of $ respect to λ.
Let $ = (F, A) be intuitionistic fuzzy soft set over U = {x1 , x2 , , · · · , xn } where
U is a set of alternatives and A = {ε1 , ε2 , · · · , εm } is a set of criteria. Assume that
the criteria are dependent to each other. If the value of F (εj )(xi ) is denoted by
αij = (µij , νij ), then the value of the ith alternative is αi = {αij |j = 1, 2, · · · , m},
i = 1, 2, · · · , n. We can establish an exact model of entropy weights for determining
weight ωj of the jth criteria εj ∈ A as follows:
ωj =
1−H(F (εj ))
,
m−Σm
k=1 H(F (εk ))
then ω = (ω1 , ω2 , · · · , ωm ) is a set of criteria weights.
Now we show our adjustable approach as follows:
Algorithm 2.
(1) Input the (result) intuitionistic fuzzy soft set $ = (F, A).
(2) Input a threshold intuitionistic fuzzy set λ : A → [0, 1] × [0, 1] for decision
making.
(3) Compute the level soft set L($; λ).
(4) Present the level soft set L($; λ) in tabular form and compute the choice
value ci of xi , ∀i.
(5) If |κL($;λ) (ΥL($;λ) )| = 1, then the optimal decision is to select xk where
xk ∈ κL($;λ) (ΥL($;λ) ), else if |κL($;λ) (ΥL($;λ) )| ≥ 2, then go to 6.
(6) Compute the entropy weight ωj of each parameter εj ∈ A.
(7) Calculate the score S(xi ) of alternative xi ∈ κL($;λ) (γ):
Pm
S(xi ) = j=1 ωj S(αij ).
Then the optimal decision is to select xk if
S(xk ) = max{S(xi )|xi ∈ κL($;λ) (γ)}.
It is see that our proposed algorithm is an adjustable one for Jiang et al.’s.
Compared with Jiang et al. algorithm, our algorithm emphasizes how to choose the
best one from the alternatives which has the largest numbers of criteria satisfied by
the alternatives at certain level of membership degrees.
5. Example illustration
To illustrate the basic idea of Algorithm 2, we apply it to the following examples.
Example 5.1. Let us reconsider Example 3.1, if we deal with this problem by
threshold intuitionistic fuzzy set λ = topbottom$ , then we obtain the level soft set
L($; topbottom$ ) with choice values with tabular representation as in Table 2. We
421
Yang et al./Ann. Fuzzy Math. Inform. 6 (2013), No. 2, 415–424
have ΥL($;λ) = 5, |κL($;topbottom$ ) (5)| = 2 and h5 , h6 ∈ κL($;topbottom$ ) (5). By
calculating the set of criteria weights
ω = (0.1805, 0.1747, 0.1890, 0.1770, 0.1371, 0.1418),
we get S(h5 ) = 0.7581 and S(h6 ) = 0.6534. So the optimal decision is to select h5 .
Example 5.2. Let us suppose there is an investment company, which wants to
invest a sum of money in the best option (adapted from [8]). Let us consider an
intuitionistic fuzzy soft set G = (F, A) which describes the “attractiveness of companies” that the investment company is considering for investment. Suppose there
are possible five alternative companies in the domain U = {u1 , u2 , u3 , u4 , u5 } : u1 is
a car company, u2 is a food company, u3 is a computer company, u4 is a arms company, u5 is a TV company. The investment company must take a decision according
to criteria set A = {a1 , a2 , a3 , a4 }, where a1 is the risk analysis, a2 is the growth
analysis, a3 is the social-political impact analysis, a4 is the environment analysis.
The five possible alternatives ui (i = 1, 2, 3, 4, 5) are to be evaluated using the intuitionistic fuzzy information by the decision maker under the above criteria, as listed
in Table 4.
If we deal with the problem with threshold intuitionistic fuzzy
P set λ = midG
where midG : A → [0, 1] × [0, 1] defined by µmidG (a) = |U1 | u∈U µF (a) (u) and
P
νmidG (a) = |U1 | u∈U νF (a) (u), thus we obtain the level soft set L(G; midG ) with
choice values with tabular as in Table 5.
Thus ΥL(midG ;λ) = 3, then |κL(G;midG ) (3)| = 1 and u5 ∈ κL(G;midG ) (3). Hence the
optimal decision is to select u5 . The result is consistent with [23].
Table 4
Intuitionistic fuzzy soft set G = (F, A)
U
a1
a2
a3
a4
u1 (0.5, 0.4) (0.6, 0.3) (0.3, 0.6) (0.6, 0.2)
u2 (0.7, 0.3) (0.7, 0.2) (0.7, 0.2) (0.4, 0.5)
u3 (0.5, 0.4) (0.5, 0.4) (0.5, 0.3) (0.4, 0.3)
u4 (0.8, 0.1) (0.6, 0.3) (0.3, 0.4) (0.2, 0.6)
u5 (0.6, 0.2) (0.8, 0.1) (0.7, 0.1) (0.7, 0.3)
Level
U
u1
u2
u3
u4
u5
soft
a1
0
0
0
1
0
Table 5
set G = (F, A) with choice values
a2 a3 a4 Choice value (ci )
0 0 1
c1 = 1
1 1 0
c2 = 2
0 1 0
c3 = 1
0 0 0
c4 = 1
1 1 1
c5 = 3
422
Yang et al./Ann. Fuzzy Math. Inform. 6 (2013), No. 2, 415–424
6. Conclusions
In the paper, through analyzing Jiang et al. approach, we proposed a novel
decision-making approach with entropy weight based on intuitionistic fuzzy soft
sets. We computed the weight of each parameter by our new entropy measure and
chose the final optimal decisions based on the scores of alternatives at certain level.
To extend this work, one can apply the entropy measure of IFSs to other practical
applications or discuss how to cope with the weights of parameters.
Acknowledgements. The authors are thankful to the anonymous referees for
their valuable comments and constructive suggestions which greatly improved the
quality of this paper.
References
[1] H. Aktaş and N. Çağman, Soft sets and soft groups, Inform. Sci. 177 (2007) 2726–2735.
[2] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986) 87–96.
[3] T. Changphas and B. Thongkam, On soft Γ-semigroups, Ann. Fuzzy Math. Inform. 4(2) (2012)
217–223.
[4] F. Feng, Soft rough sets applied to multicriteria group decision making, Ann. Fuzzy Math.
Inform. 2(1) (2011) 69–80.
[5] F. Feng, Y. Li and N. Çağman, Generalized uni-int decision making schemes based on choice
value soft sets, European J. Oper. Res. 220 (2012) 162–170.
[6] F. Feng, Y. B. Jun, X. Liu and L. Li, An adjustable approach to fuzzy soft set based decision
making, J. Comput. Appl. Math. 234 (2010) 10–20.
[7] F. Feng, Y. Li and V. Leoreanu-Fotea, Application of level soft sets in decision making based
on interval-valued fuzzy soft sets, Comput. Math. Appl. 60 (2010) 1756–1767.
[8] F. Herrera, E. Herrera-Viedma and L. Martnez, A fusion approach for managing multigranularity linguistic term sets in decision making, Fuzzy Sets and Systems 114 (2000) 43–58.
[9] Y. Jiang, Y. Tang and Q. Chen, An adjustable approach to intuitionistic fuzzy soft sets based
decision making, Appl. Math. Model. 35 (2011) 824–836.
[10] Y. B. Jun and C. H. Park, Applications of soft sets in ideal theory of BCK/BCI-algebras,
Inform. Sci. 178 (2008) 2466–2475.
[11] Y. B. Jun, K. J. Lee and E. H. Roh, Intersectional soft BCK/BCI-ideals, Ann. Fuzzy Math.
Inform. 4(1) (2012) 1–7.
[12] Y. B. Jun, K. J. Lee and C. H. Park, Soft set theory applied to ideals in d-algebras, Comput.
Math. Appl. 57 (2009) 367–378.
[13] Z. Kong, L. Wang and Z. Wu, Application of fuzzy soft set in decision making problems based
on grey theory, J. Comput. Appl. Math. 236 (2011) 1521–1530.
[14] J. Li, G. Deng, H. Li and W. Zeng, The relationship between similarity measure and entropy
of intuitionistic fuzzy sets, Inform. Sci. 188 (2012) 314–321.
[15] P. K. Maji, R. Biswas and A. R. Roy, Fuzzy soft sets, J. Fuzzy Math. 9(3) (2001) 589–602.
[16] P. K. Maji, A. R. Roy and R. Biswas, On intuitionistic fuzzy soft sets, J. Fuzzy Math. 12
(2004) 669–683.
[17] P. K. Maji, A neutrosophic soft set approach to a decision making problem, Ann. Fuzzy Math.
Inform. 3(2) (2012) 313–319.
[18] D. Molodtsov, Soft set theory-first results, Comput. Math. Appl. 37 (1999) 19–31.
[19] A. R. Roy and P. K. Maji, A fuzzy soft set theoretic approach to decision making problems,
J. Comput. Appl. Math. 203 (2007) 412–418.
[20] M. Shabir and A. Ahmad, On soft ternary semigroups, Ann. Fuzzy Math. Inform. 3(1) (2012)
39–59.
[21] E. Szmidt and J. Kacprzyk, Entropy for intuitionistic fuzzy sets, Fuzzy Sets and Systems 118
(2001) 467–477.
423
Yang et al./Ann. Fuzzy Math. Inform. 6 (2013), No. 2, 415–424
[22] I. K. Vlachos and G. D. Sergiadis, Intuitionistic fuzzy information-Applications to pattern
recognition, Pattern Recogn. Lett. 28 (2007) 197–206.
[23] G. W. Wei, Gray relation alanalysis method for intuitionistic fuzzy multiple attribute decision
making, Expert Syst. Appl. 38 (2011) 11671–11677.
[24] H. Wu and J. Zhan, The characterizations of some kinds of soft hemirings, Ann. Fuzzy Math.
Inform. 3(2)(2012) 267–280.
[25] Z. Xiao, K. Gong and Y. Zou, A combined forecasting approach based on fuzzy soft sets, J.
Comput. Appl. Math. 228 (2009) 326–333.
[26] Y. Yang, X. Tan and C. Meng, The multi-fuzzy soft set and its application in decision making,
Appl. Math. Model. 37 (2013) 4915–4923.
[27] J. Zhan and Y. B. Jun, Soft BL-algebras based on fuzzy sets, Comput. Math. Appl. 59 (2010)
2037–2046.
[28] L. Zhang, X. Xin and Q. Liu, Algebra properties and fuzzy entropy of vague sets (in chinese),
Computer Engineering and Applications 46 (2010) 32–35.
Yong Wei Yang ([email protected])
Department of Mathematics, Northwest University, Xi’an, 710127, China
Ting Qian ([email protected])
Department of Mathematics, Northwest University, Xi’an, 710127, China
424